Setting aside paradoxical linkages such as Bennett’s, Bricard’s or Goldberg’s, the mobility of single loop linkages seemed, with the developments on mobility analysis carried out in the last five years, a closed chapter in kinematic research. However, recent developments on the mobility of parallel platforms have shed additional insight into the problem. This contribution attempts to unify the results obtained in the last five years in the area of mobility of single-loop kinematic chains to state what appears to be a final word on the subject.

This paper reformulates and extends the new, group theoretic, mobility criterion recently developed by the authors, Rico and Ravani [1]. In contrast to the Kutzbach-Gru¨bler criterion, the new mobility criterion, and the approach presented apply to a large class of overconstrained linkages. The criterion is reformulated, in terms of the well known Jacobian matrices, for exceptional linkages; it is extended to linkages with partitioned mobility as well as trivial linkages. In addition, an extension of the criterion is presented that would allow the computation of degrees-of-freedom of several cases of paradoxical linkages. The case of classical paradoxical linkages such as the Bennett and Goldberg linkages still remains unsolved but some insight into the application of the new mobility criterion for these linkages is also presented.

Recent results have shown that the application of group theory to the Euclidean group, E (3), and its subgroups yields a new and improved mobility criterion. Unlike the well known Kutzbach-Gru¨bler criterion, this improved mobility criterion yields correct results for both trivial and exceptional linkages. Unfortunately, this improved mobility criterion requires a little bit more than counting links and kinematic pairs. An important advance was made when it was proved that the improved mobility criterion, originally stated in a language of group theory and subsets and subgroups of the Euclidean group, E(3), can be translated into a language of the Lie algebra, e (3), of the Euclidean group, E (3), and its vector subspaces and its subalgebras. The language of the Lie algebra, e (3), is far simpler than the nonlinear language of the Euclidean group, E (3). Still, the computations required for the improved mobility criterion are more involved than those required for the Kutzbach-Gru¨bler criterion, and it might preclude the employment of the improved mobility criterion in prospective tasks such as the number synthesis of parallel and modular manipulators. This contribution dispels these doubts by showing that the improved criterion can be easily implemented by a simple computer program. Several examples are included.

This paper presents a generalization of Kutzbach-Gru¨bler criterion for mobility analysis of kinematic chains based on group theory. This new mobility criterion applies to exceptional linkages and reduces to a group theoretic representation of Kutzbach-Gru¨bler criterion for trivial linkages. Furthermore, it is shown that using sets and groups of Euclidean displacements, one can distinguish between trivial, exceptional, and paradoxical linkages. Using these concepts, formal definitions of trivial, exceptional, and paradoxical linkages are presented and it is shown that there are two classes of paradoxical linkages. In addition, a new class of linkages is identified that have partitioned mobility. This work builds upon and extends the work of Herve´ and Fanghella and Galletti in application of group theory to analysis of kinematic chains.

It is well known that the Kutzbach-Gru¨bler criterion fails in computing the mobility of exceptional and paradoxical linkages, these two classes of linkages are sometimes referred to as overconstrained. Recently, the authors [1] developed a new mobility criterion that correctly computes the mobility of trivial and exceptional linkages. In the form presented in [1; 2] the mobility criterion can not compute the mobility of paradoxical linkages. This paper presents an extension of the mobility criterion developed by the authors that allows the computation of the degrees-of-freedom of large classes of paradoxical linkages, the conditions for taking advantage of this extension are revealed. Moreover, a hypothesis for applying this extension to classical paradoxical linkages such as the Bennett and Goldberg linkages is presented. Several examples are used to illustrate the method.